Problem: Is ${541789}$ divisible by $9$ ?
A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {541789}= &&{5}\cdot100000+ \\&&{4}\cdot10000+ \\&&{1}\cdot1000+ \\&&{7}\cdot100+ \\&&{8}\cdot10+ \\&&{9}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {541789}= &&{5}(99999+1)+ \\&&{4}(9999+1)+ \\&&{1}(999+1)+ \\&&{7}(99+1)+ \\&&{8}(9+1)+ \\&&{9} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {541789}= &&\gray{5\cdot99999}+ \\&&\gray{4\cdot9999}+ \\&&\gray{1\cdot999}+ \\&&\gray{7\cdot99}+ \\&&\gray{8\cdot9}+ \\&& {5}+{4}+{1}+{7}+{8}+{9} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${541789}$ is divisible by $9$ if ${ 5}+{4}+{1}+{7}+{8}+{9}$ is divisible by $9$ Add the digits of ${541789}$ $ {5}+{4}+{1}+{7}+{8}+{9} = {34} $ If ${34}$ is divisible by $9$ , then ${541789}$ must also be divisible by $9$ ${34}$ is not divisible by $9$, therefore ${541789}$ must not be divisible by $9$.